Three-fast-searchable graphs

In the edge searching problem, searchers move from vertex to vertex in a graph to capture an invisible, fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which, at every move, a new edge of the graph G must be guaranteed to be free of the intruder. That is, once all searchers are placed the graph G is cleared in exactly |E(G)| moves. Such a restriction obviously necessitates a larger number of searchers. We examine this model, and characterize graphs for which 2 or 3 searchers are sufficient. We prove that the corresponding decision problem is NP-complete.